Sampling measures for the Gabor transform

Sampling measures of a space HH are measures μμ such that ∫|f|pdμ≈‖f‖p for every function f∈Hf∈H. Here we present a study of these measures for the specific case of model spaces of the Gabor transform. These spaces are the continuous transforms VφgVφg of functions gg in a modulation space Mp,qMp,q with respect to a fixed window φ∈M1φ∈M1, the Feichtinger Algebra.We obtain a characterization of these measures in terms of discrete sampling sets and stability results. For a special class of windows that includes most of the important examples of Gabor atoms we also find sufficient conditions in terms of weak limits of uniqueness sets. We also discuss some applications.